Diabatic description of charmoniumlike mesons. II. Mass corrections and strong decay widths

Diabatic description of charmoniumlike mesons. II. Mass corrections and strong decay widths R. Bruschini 1,* and P. González 1,2,† 1Unidad Teórica, Instituto de Física Corpuscular (Universidad de Valencia–CSIC), E-46980 Paterna (Valencia), Spain 2Departamento de Física Teórica, Universidad de Valencia, E-46100 Burjassot (Valencia), Spain (Received 13 January 2021; accepted 10 March 2021; published 13 April 2021) From a diabatic bound state approach to JPC ¼1−− and ð0;1;2Þþþ charmoniumlike resonances below 4.1 GeV, formulated in terms of c¯ cand closed meson-meson channels, we calculate mass shifts and widths due to open meson-meson channels. This calculation does not involve any new free parameter, so comparison of our predictions with existing data provides a direct test of our approach. Further mass corrections are also estimated and good agreement with the measured masses comes out. As for the calculated widths, overall reasonable, they point out to the need of some refinement of our current bound state approximation for an accurate description of data. These results give additional support to the diabatic approach in QCD as an adequate framework for a complete unified description of conventional and unconventional charmoniumlike resonances. In this respect, the experimental discovery of a predicted 2þþ resonance with a mass around 4 GeV would be of special relevance. DOI: 10.1103/PhysRevD.103.074009 I. INTRODUCTION A current challenge in hadron physics is to achieve a QCD based description of (unconventional) quarkoniumlike mesons such as χc1ð3872Þ,Xð3915Þ, and others [1] discovered in the past two decades, whose properties do not correspond to a very dominant conventional heavy quark (Q)–heavy antiquark ð¯ QÞmeson structure. Although there are nowadays compelling experimental indications of the presence of additional open-flavor mesonmeson components in some of these unconventional states, the QCD theoretical implementation of such components, together with the Q¯ Qones for a complete quarkoniumlike description, presents some difficulties (for a comprehensive review of the experimental and theoretical situation, see [2–14] and references therein). Just recently, noticeable progress in this direction, based on lattice QCD results, has been reported. More concretely, unquenched lattice QCD calculations of the energies for static Qand ¯ Qsources, when the Q¯ Qconfiguration mixes with one or two open-flavor meson-meson configurations, have been carried out [15,16]. These static energies can be directly related to the potential matrix entering in a multichannel Schrödinger equation for the Q¯ Qand meson-meson components of a quarkoniumlike meson. This relation has been used to explore the bottomoniumlike spectrum when only one (average) meson-meson channel is taken into account [17]. A more general and systematic formulation of this relation for any number of meson-meson channels has been recently proposed [18]. This formulation translates the diabatic approach used in molecular physics to tackle the electronic configuration mixingproblem[19]tothestudyofquarkoniumlikesystems. The direct diabatic correspondence between lattice QCD calculations and the potential matrix entering the Schrödinger equation constitutes a substantial improvement over existing descriptions of quarkoniumlike mesons where the interaction between Q¯ Qand meson-meson degrees of freedom has no clear connection with QCD. It is worth emphasizing that the diabatic approach goes beyond the (single-channel) Born-Oppenheimer approximation in QCD developed in [20] and applied in [21] to the description of quarkonium and quarkonium hybrids. Indeed, the diabatic approach reduces to the Born-Oppenheimer approximation for conventional quarkonium Q¯ Qforenergies far below the lowest open-flavor meson-meson threshold, but unlike the Born-Oppenheimer approximation it maintains its validity for energies close below or above that threshold. We refer to the previous paper of this series [18] for a more detailed explanation about the advantages of the diabatic approach over the Born-Oppenheimer approximation. Therefore, the so-called diabatic approach in QCD provides an appropriate framework for a unified and *[email protected].es †[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 103, 074009 (2021) 2470-0010=2021=103(7)=074009(13) 074009-1 Published by the American Physical Society complete nonperturbative description of conventional quarkonium and quarkoniumlike mesons made of Q¯ Qand meson-meson components. In the first practical application of the diabatic formalism to the description of charmoniumlike mesons we have considered them stable against decays into open-flavor meson-meson channels, neglecting the fact that states above any possiblemeson-meson threshold with the same quantum numbers are unstable. This approximation has been implemented by solving the multichannel Schrödinger equation taking into account that a bound state solution with a correct asymptotic behavior requires that meson-meson components with thresholds below the mass of the bound state be neglected. This description, which has been applied to isoscalar I¼0states with JPC ¼1−− and ð0;1;2Þþþ below 4.1 GeV, is expected tobewell suited for narrow states as the ones below the first absolute meson-meson threshold D¯ D,or thefirstexcited1þþ state,whichisassignedtoχc1ð3872Þ,for which there is no coupled lower threshold. For the other statesthetreatmentmayalsobesuitedifmasscorrectionsdue to the neglected meson-meson components and widths can be properly incorporated. In this article we go a step further in the diabatic description of quarkoniumlike mesons by evaluating, from the calculated bound state solutions, the contribution of the neglected meson-meson components to their masses and decay widths. Then, a more detailed comparison with data may be done. In this regard, one should always keep in mind that there might still exist experimental quarkoniumlike resonances escaping from the bound state approximation we follow. The contents of the article are organized as follows. In Sec. II a brief resume of the diabatic bound state description of quarkoniumlike mesons is presented. In Sec. III we detail the framework for the calculation of mass corrections and widths to open-flavor meson-meson channels. The results obtained for charmoniumlike systems are listed and discussed in Sec. IV. Finally, in Sec. Vwe summarize our main results and conclusions. II. DIABATIC BOUND STATE DESCRIPTION OF QUARKONIUMLIKE MESONS The diabatic approach in QCD has been detailed in [18]. Quarkoniumlike mesons, characterized by quantum numbers JPC, are assigned to solutions of the multichannel Schrödinger equation, ðKþVðrÞÞΨðrÞ¼EΨðrÞ;ð1Þ where ΨðrÞis a column vector notation for the wave function, ΨðrÞ¼ 0 B B B B B B @ ψQ¯ QðrÞ ψð1Þ M¯ MðrÞ . . . ψðNÞ M¯ MðrÞ 1 C C C C C C A ;ð2Þ with ψðiÞ M¯ MðrÞ,i¼1;2…standing for the ith meson-meson component. K is the kinetic energy matrix K¼ 0 B B B B B B B B @ −1 2μQ¯ Q ∇2 −1 2μð1Þ M¯ M ∇2 .. . −1 2μðNÞ M¯ M ∇2 1 C C C C C C C C A ;ð3Þ where μðiÞ M¯ Mis the reduced mass of the ith meson-meson component, and matrix elements equal to zero are not displayed. VðrÞis the diabatic potential matrix VðrÞ¼ 0 B B B B B B @ VCðrÞVð1Þ mixðrÞ  VðNÞ mixðrÞ Vð1Þ mixðrÞTð1Þ M¯ M . . ... . VðNÞ mixðrÞTðNÞ M¯ M 1 C C C C C C A :ð4Þ The diagonal term VCðrÞstands for the Q¯ QCornell potential VCðrÞ¼σr−χ rþmQþm¯ Q−βð5Þ with σ,χ,mQand βbeing the string tension, the color Coulomb strength, the heavy quark mass, and a constant fixing the origin of the potential respectively. Any of the other diagonal terms stands for the mass of a meson-meson threshold TðiÞ M¯ M¼mðiÞ MþmðiÞ ¯ Mð6Þ with mðiÞ Mand mðiÞ ¯ Mbeing the masses of the corresponding meson and antimeson. The off-diagonal term VðiÞ mixðrÞstands for the mixing potential between the Q¯ Qand the ith meson-meson component, which can be calculated ab initio from lattice QCD. Notice that no interaction potential between different meson-meson components is considered, what it is justified for isolated, well separated meson-meson thresholds with R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 074009 (2021) 074009-2 no overlap at all (including the meson widths). Notice though that an indirect interaction through their coupling to the Q¯ Qchannel is present. Moreover, for the spherically symmetric and spinindependent diabatic potential matrix we use, each Q¯ Q configuration with a distinct value of ðlQ¯ Q;s Q¯ QÞis taken as a channel per se, and the same for each meson-meson configuration with a distinct value of ðlðiÞ M¯ M;s ðiÞ M¯ MÞ. The possible values of ðl; sÞare determined by the JPC quantum numbers of the quarkoniumlike system under consideration. Then, for a given JPC the mixing potential entering the coupled radial equations, VðiÞ mixðrÞ, is the same between any one of the ðlQ¯ Q;s Q¯ QÞand any one of the ðlðiÞ M¯ M;s ðiÞ M¯ MÞ channels. In order to calculate bound states some constraints are imposed. First, a finite number of thresholds is considered. This is justified because for a given bound state the probability of meson-meson components corresponding to thresholds far above the mass of the bound state is negligible. Additionally, only meson-meson components above the bound state mass are considered. Otherwise the free-wave behavior introduced by meson-meson components with threshold below the mass of the bound state would prevent us from obtaining a physical (bound state) solution. In practice this is done by selecting a subset of thresholds, from a given threshold up, and calculating the bound states below it. In order to avoid a possible double counting when different threshold selections are considered a one to one correspondence between the Cornell Q¯ Qand the quarkoniumlike bound states is assumed. Notice that a solution of the Schrödinger equation above threshold is possible, but the eigenstates would represent meson-meson scattering states. In such a context quarkoniumlike mesons would be described as resonances in a properly defined meson-meson scattering problem. A development of a diabatic description of the coupled channel meson-meson scattering, out of the scope of the present article, is under way. The diabatic bound state approach has been applied to isoscalar I¼0charmoniumlike (Q¼c) mesons with JPC ¼1−− and ð0;1;2Þþþ below 4.1 GeV for which the threshold-threshold interactions can be safely neglected. For the Cornell potential (5), standard values of the parameters σ¼925.6MeV=fm;ð7aÞ χ¼102.6MeVfm;ð7bÞ mc¼1840 MeV;ð7cÞ β¼855 MeV ð7dÞ have been chosen [22]. The meson-meson thresholds, calculated from the masses of charmed and charmed strange mesons in [1], are listed in Table I. It is worth remarking that the use of the masses of the experimental thresholds introduces some implicit spin dependence in the description. The possible values of lc¯ cand lðiÞ M¯ Mcontributing to a given set of quantum numbers JPC are shown in Table II where the common notation DðsÞto refer to charmed as well as to charmed strange mesons is used. As for the mixing potential, the current dearth of unquenched lattice data for charmoniumlike systems [23,24] prevents its ab initio calculation. Instead, its form has been parametrized from lattice results for bottomoniumlike systems [15,16]. The reasons for this parametrization have been detailed in [18]. It reads jVðiÞ mixðrÞj ¼ Δ 2exp−ðVCðrÞ−TðiÞ M¯ MÞ2 2σ2ρ2ð8Þ or equivalently, at distances for which VCðrÞ≈ σrþmQþm¯ Q−β, jVðiÞ mixðrÞj ≈ Δ 2exp−ðr−rðiÞ cÞ2 2ρ2ð9Þ where Δ 2stands for the strength and ρ, the width of the second Gaussian, for the radial scale of the mixing, whereas rðiÞ cdenotes the crossing radius defined by VCðrðiÞ cÞ¼TðiÞ M¯ M:ð10Þ TABLE I. Low-lying open-charm meson-meson thresholds. Threshold masses TðiÞ M¯ Mfrom the measured charmed and charmed strange meson masses [1]. M¯ MTM¯ M(MeV) D¯ D3730 D¯ Dð2007Þ3872 Dþ sD− s3937 Dð2007Þ¯ Dð2007Þ4014 Dþ sD− s4080 TABLE II. Values of lc¯ cand lðiÞ M¯ Mcorresponding to definite values of JPC. A missing entry for lðiÞ M¯ Mmeans that the particular meson-meson configuration cannot form a state with the corresponding quantum numbers. JPC lc¯ clDðsÞ¯ DðsÞlDðsÞ¯ D ðsÞlD ðsÞ¯ D ðsÞ 0þþ 1 0 0, 2 1þþ 1 0,2 2 2þþ 1, 3 2 2 0, 2, 4 1−− 0, 2 1 1 1, 3 DIABATIC …II. MASS CORRECTIONS …PHYS. REV. D 103, 074009 (2021) 074009-3 Let us realize that the mixing is only effective in an interval around rðiÞ cdetermined by the value of ρ. The numerical values of ρand Δ, Δ¼130 MeV;ð11aÞ ρ¼0.3fm;ð11bÞ were obtained in [18] by fitting the mass of χc1ð3872Þvia a calculation involving only the D¯ Dthreshold. In the present calculation we have also included the tiny effect of the D¯ Dthreshold, therefore the use of the same values of the parameters gives rise to a very little discrepancy between the calculated mass of χc1ð3872Þand its measured value. The calculated spectrum of ð0;1;2Þþþ states, containing one c¯ ccomponent with lc¯ c¼1(1p)ortwoc¯ ccomponents with lc¯c¼1,3(2pand 1f) and 1−− states, containing one c¯ ccomponent with lc¯ c¼0(1s)ortwoc¯ ccomponents with lc¯ c¼0,2(2sand 1d) is shown in Table III. Let us note that the content of Table III differs from the one of Tables V and VI in [18] due to some technical improvement in the numerical calculation, to the inclusion of lc¯ c¼3and lðiÞ M¯ M¼4channels which allows for the prediction of a new 2þþ resonance (see below), and to the exclusion of the 1−− excited state assigned to ψð4040Þ. The reason for this exclusion is that this state is peculiar in the sense that the improved calculation puts the state below the D¯ Dthreshold preventing the decay to any D¯ D channel. However, the mass correction, calculated as explained in the next section, makes the mass to be so close to the D¯ Dthreshold that a small variation in the parameters (well within our expected uncertainty) puts the state above threshold permitting such decay as experimentally observed. This instability suggests that a refinement of our approximation is necessary for an appropriate description of ψð4040Þ. This refinement may require a more complete treatment incorporating all meson-meson channels on the same foot (as in a diabatic coupled mesonmeson scattering framework not yet developed), and interactions between the bound states through their couplings to the same thresholds. It may also have to do with an improvement of the diabatic potential, through the inclusion of spin dependent terms and the consideration of a differentiated mixing for hidden strange thresholds. For a meaningful comparison with data we have to take into account that our Cornell potential does not contain spin-dependent terms. Then, for pure ðnlÞc¯ cstates the calculated masses have to be compared to the ðnlÞ experimental centroids; in the other cases, where mesonmeson components are present, since they are specific for a set of JPC quantum numbers, the comparison has to be done with the experimental candidates with the same JPC. As can be checked the calculated masses for the ð0;1;2Þþþ ground states, which are practically pure charmonium states, are pretty close to the experimental ground state centroid from 1PJstates at 3525.30 0.11 MeV. As for the calculated masses of the first excited ð0;1;2Þþþ states, all containing significant meson-meson components, can be put in good correspondence with the existing experimental candidates. So, χc1ð3872Þ, with a measured mass of 3871.69 0.17 MeV has been assumed to correspond to the first excited 1þþ state, which implies that it is essentially ð95%ÞaD¯ Dbound state; χc0ð3860Þ, with a measured mass of 3862þ26þ40 −32−13 MeV, corresponds to the first excited 0þþ state with an important ð35%ÞDs¯ Ds component; and χc2ð3930Þwith a measured mass of 3922.21.0MeV corresponds to the first excited 2þþ state with small but non-negligible ð7%ÞDs¯ Dsand ð13%Þ D¯ Dcomponents. It should also be remarked that the calculated masses for the first excited ð0;2Þþþ states are in complete agreement with data regarding their positions with respect to the Ds¯ Dsthreshold, both below it. There is an additional Jþþ experimental resonance below 4.1 GeV, the Xð3915Þ, with a measured mass of 3918.41.9MeV, whose quantum numbers are not well TABLE III. Masses, c¯ cprobabilities, and meson-meson probabilities, for ð0;1;2Þþþ and 1−− charmoniumlike states calculated neglecting interaction with open thresholds. The c¯ cprobabilities from different values of lc¯ c(see Table II) have been reported inside parentheses. A missing entry under a meson-meson configuration means that the corresponding component gives negligible (i.e., probability inferior to 1%) or no contribution at all to the state. JPC Mass (MeV) c¯ cD¯ DD ¯ DDs¯ DsD¯ DDs¯ D sD s¯ D s 0þþ 3508.8 99% 1% 3918.9 60% 35% 5% 1þþ 3509.8 100% 3871.5 5% 95% 2þþ 3508.7 (100, 0) % 3909.0 (69, 8) % 7% 13% 1% 2% 4006.6 (18, 53) % 28% 1% 1−− 3082.4 (100, 0) % 3658.8 (92, 1) % 4% 1% 1% 1% 3785.8 (1, 95) % 2% 1% 1% R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 074009 (2021) 074009-4 established being possible a 0þþ as well as a 2þþ assignment. In [18] we tentatively identified it with χc0ð3860Þbut observed through the (J=ψω) decay channel. However, a 2þþ identification with χc2ð3930Þis also feasible. In this regard, the quantitative analysis of the decay widths to open-flavor meson-meson channels may shed some light on its correct assignment. There is also an additional theoretically predicted resonance corresponding to the second 2þþ excited state with no existing experimental candidate to be compared with. This prediction is quite robust in the sense that it comes out from the presence of the 1fCornell state at 4034 MeV. Hence, the experimental finding of this resonance would provide a strong support to our treatment. In this respect, the analysis of its dominant strong decay widths that we carry out later on may guide experimental searches. As for 1−− a good correspondence of the calculated masses with data can be established as well. So, the ground state is a conventional ð100%Þc¯ ccharmonium 1sstate, its calculated mass being pretty close to the experimental 1s centroid at 3068.65 0.13 MeV [from J=ψand ηcð1sÞ]. For the first excited state, containing a very big ð92%Þ2s c¯ ccomponent and a quite small ð4%ÞD¯ Dcomponent, the calculated mass is close to the measured mass of ψð2sÞ: 3686.097 0.010 MeV. The second excited state is very dominantly a 1dc ¯ cstate ð95%Þ, its calculated mass comparing well with the mass of ψð3770Þ:3778.1 0.7MeV. It is worth pointing out that we expect the calculated wave functions and masses for states with a nonvanishing Ds¯ Dsand Ds¯ D sprobability to be more uncertain that the ones calculated for states with no such components. The reason is that the parameters of the mixing potential have been fixed from the D¯ Dthreshold [through the fitting of the mass of χc1ð3872Þ]. Then, although the use of the same parameters for the D¯ Dand D¯ Dthresholds, differing only from D¯ Din their spins, is justified in our spin independent treatment, it may not be the case for Ds¯ Dsand Ds¯ D s.For these thresholds the interaction with c¯ cinvolves the creation of a strange quark-antiquark pair so that the strength and radial scale for the mixing may differ from the one associated to the light quark ðu; dÞpair creation involved in the D¯ D-c¯ cinteraction. Unfortunately, the lack of detailed data clearly indicating the presence of Ds¯ Dsand Ds¯ D scomponents in some experimental resonance prevents us from fixing the specific mixing parameters for these thresholds and from estimating quantitatively the uncertainty deriving from using the same parameters as for the other thresholds. Notwithstanding this some qualitative prescription can be formulated in the sense that the bound state masses and wave functions are better described if not involving Ds¯ Dsand Ds¯ D scomponents or involving a very small percentage of them. Keeping this in mind, the reasonable agreement of the calculated masses with existing data for all the excited states suggests that the diabatic bound state approximation to charmoniumlike mesons, below and above threshold, may be appropriate and constitutes a good starting point for the evaluation of mass corrections and widths. Next we center on the mass shifts and widths having to do with the neglected thresholds. As we shall prove they cannot be calculated using perturbation theory against our previous assumption in [18]. III. MASS CORRECTIONS AND WIDTHS In order to evaluate the effect of the nneglected thresholds in the calculation of bound states with masses above these thresholds, let us start from the Hamiltonian, H0¼KþV0;ð12Þ whose matrix representation in configuration space is given by the sum of K from (3) and V0ðrÞ¼ 0 B B B B B B B B B B B B B B B B @ VCðrÞVðnþ1Þ mix ðrÞ…VðNÞ mixðrÞ Tð1Þ M¯ M .. . TðnÞ M¯ M Vðnþ1Þ mix ðrÞTðnþ1Þ M¯ M . . ... . VðNÞ mixðrÞTðNÞ M¯ M 1 C C C C C C C C C C C C C C C C A :ð13Þ DIABATIC …II. MASS CORRECTIONS …PHYS. REV. D 103, 074009 (2021) 074009-5 The wave functions, corresponding to solutions of the Schrödinger equation H0jΨ0i¼E0jΨ0ið14Þ can be expressed as Ψ0ðrÞ¼ 0 B B B B B B B B B B B B B B B @ ψQ¯ QðrÞ ˜ ψð1Þ M¯ MðrÞ . . . ˜ ψðnÞ M¯ MðrÞ ψðnþ1Þ M¯ MðrÞ . . . ψðNÞ M¯ MðrÞ 1 C C C C C C C C C C C C C C C A ;ð15Þ where we have used the notation ˜ ψM¯ Mto indicate the noncoupled (free) meson-meson components. Let us realize that any bound state obtained when neglecting the thresholds with j¼1;…;n corresponds to a solution of (14) of the type ΨðkÞ boundðN−nÞðrÞ¼ 0 B B B B B B B B B B B B B @ ψðkÞ Q¯ QðrÞ ψðnþ1ÞðkÞ M¯ MðrÞ . . . ψðNÞðkÞ M¯ MðrÞ 1 C C C C C C C C C C C C C A ð16Þ with eigenvalue MðkÞ ðN−nÞgiven by the calculated bound state mass, where kstands for a set of quantum numbers labeling the bound state. On the other hand the free meson-meson states for any of the meson-meson pairs in the channels 1;…;ncorrespond to solutions of (14) of the type Ψðp;sÞ freeðjÞðrÞ¼ 0 B B B B B @ ˜ ψðjÞðp;sÞ M¯ MðrÞ1 C C C C C A ð17Þ with eigenvalues EðpÞ ðjÞ¼TðjÞ M¯ Mþp2 2μðjÞ M¯ M , with pthe relative meson-meson momentum and sthe total meson-meson spin. Notice that there is a continuum of eigenstates for each value of j. The interaction Hamiltonian between bound and free states is provided by those mixing terms in the diabatic potential matrix Vthat have been discarded in the construction of V0: HI≡V−V0:ð18Þ Its matrix representation in configuration space reads HIðrÞ¼ 0 B B B B B B B B B B @ Vð1Þ mixðrÞ  VðnÞ mixðrÞ Vð1Þ mixðrÞ . . . VðnÞ mixðrÞ 1 C C C C C C C C C C A ; ð19Þ where it can be seen that HIcouples the Q¯ Qcomponent ψQ¯ QðrÞof the bound state and the free meson-meson channels ˜ ψðjÞ M¯ MðrÞwith j¼1;…;n. In order to study the effect of HIwe have to realize that the exact coincidence of the energy (mass) of the bound state with one energy of the continuum for any of the MðjÞ¯ MðjÞcomponents makes the ordinary perturbation theory inadequate (see the Appendix for a detailed explanation). Instead, a procedure based on the solution of the Schrödinger equation for H0þHIhas been developed in atomic and nuclear physics, see for instance [25], and in hadronic spectroscopy [26]. In general, for a given set of quantum numbers JPC, this procedure involves the diagonalization of a matrix containing the interactions between bound states induced through their coupling to the same meson-meson states. However, as detailed in what follows the bound states we start from have been calculated from Hamiltonians involving different sets of neglected thresholds. Hence we can only do a consistent calculation if the meson-meson channel induced interactions between different bound states are not taken into account. Then, the problem gets reduced to the calculation of the corrections to one bound state due to its interaction with the specific neglected thresholds. To be more precise let us consider Q¼c.ForJPC ¼ 0þþ there is one bound (ground) state when no thresholds are neglected and two bound states (ground and first excited) when the threshold D¯ Dis neglected. The ground state is 100% c¯ cin both cases which means that it is decoupled from any threshold. Hence, we restrict our study to one 0þþ (first excited) bound state interacting with the D¯ Dthreshold. Similarly, for the first excited 2þþ state obtained when neglecting the D¯ Dand D¯ Dthresholds, we R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 074009 (2021) 074009-6 calculate the corrections to the 2þþ (first excited) bound state interacting with D¯ Dand D¯ D. As for the second excited 2þþ state, obtained when neglecting the D¯ D,D¯ D and Ds¯ Dsthresholds, we calculate its corrections due to these thresholds forgetting about its interaction, induced through its coupling to D¯ D, with the first excited bound state obtained with the same Hamiltonian (notice that this first excited bound state differs from the previous one calculated without neglecting Ds¯ Ds). For the 1−− states we follow exactly the same protocol. So, for the second excited state we calculate its corrections due to D¯ D. Having reduced the calculation of the corrections to the one bound state case, the physical effect of the coupling to the continuum is to dilute the bound state through a band of stationary scattering states. The expressions for the mass and width of the resulting resonance can be found in [25,26]. Let us detail them for one bound state and one threshold. Let jAibe the bound state with quantum numbers JPC, and jBCia free meson-meson state. In the center of mass frame the momentum of the resonance vanishes and EB¼mBþp2 BC 2mB ð20aÞ EC¼mCþp2 BC 2mC ð20bÞ with p2 BC ¼2μBCðEBC −mB−mCÞð21Þ and EBC ≡EBþEC. Then the mass correction is written as M−MA¼X msB msC; PZdpBC jhpBC;sB;m sB;s C;m sCjHIjAij2 M−EBC ; ð22Þ where M is the mass of the resonance, HIstands for the interaction Hamiltonian operator, and PRindicates the Cauchy principal value integral. To calculate the matrix element in the numerator we manipulate it as hpBC;sB;m sB;s C;m sCjHIjAi ¼X s;ms CmsB;msC;ms sB;sC;s hpBC;sB;s C;s;m sjHIjAi:ð23Þ The wave function for the meson-meson state reads hpBC;sB;s C;s;m sjri ¼e−ipBC·r ð2πÞ3 2 ξms† s ¼X l;ml ffiffiffi 2 π ri−ljlðpBCrÞYml lðˆ pBCÞYml lðˆ rÞξms† s ¼X l;ml ffiffiffi 2 π ri−ljlðpBCrÞYml lðˆ pBCÞ ×X J0;m0 J Cml;ms;m0 J l;s;J0YJ0;m0 J† l;s ðˆ rÞ;ð24Þ where we have defined YJ0;m0 J l;s ðˆ rÞ¼X m0 l;m0 s Cm0 l;m0 s;m0 J l;s;J Ym0 l lðˆ rÞξm0 s s:ð25Þ Then as the interaction Hamiltonian only connects the Q¯ Q component ψQ¯ QðrÞof the bound state Aand the free BC state, both with the same J,mJquantum numbers, we get hpBC;sB;s C;s;m sjHIjAi ¼X l;ml Cml;ms;mJ l;s;J Yml lðˆ pBCÞIlðpBCÞ;ð26Þ where we have defined IlðpBCÞ≡ffiffiffi 2 π ri−lZdrr2jlðpBCrÞVðBCÞ mix ðrÞuQ¯ QðrÞð27Þ with uQ¯ Qthe sum of the Q¯ Qradial wave functions for the several ðlQ¯ Q;s Q¯ QÞchannels in the bound state A. Hence we obtain hpBC;sB;m sB;s C;m sCjHIjAi ¼X s;ms;l;ml CmsB;msC;ms sB;sC;s Cml;ms;mJ l;s;J Yml lðˆ pBCÞIlðpBCÞ:ð28Þ By substituting this in (22), integrating in spherical coordinates, and using angular momentum algebra it is easy to show that M−MA¼PZdpBC p2 BC M−EBC X l;s jIlðpBCÞj2:ð29Þ We can go further integrating over EBC instead of over pBC using dEBC dpBC ¼pBC mB þpBC mC ¼pBC μBC ;ð30Þ which gives DIABATIC …II. MASS CORRECTIONS …PHYS. REV. D 103, 074009 (2021) 074009-7 pBCdpBC ¼μBCdEBC:ð31Þ Thus we obtain M−MA¼PZdEBCμBC pBC M−EBC X l;s jIlðpBCÞj2;ð32Þ where pBC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μBCðEBC −mB−mCÞ pð33Þ and the sum runs over the possible values land sof BC that couple to JPC. On the other hand the expression of the width reads Γ 2¼πμBCpBCX l;s jIlðpÞBCj2EBC¼M :ð34Þ The generalization of these expressions to the case of one bound state Aand nthresholds with masses below MAcan be done straightforwardly. So M−MA¼X j;lðjÞ;sðjÞ PZdEðjÞμBC pðjÞ M−EðjÞ jIlðjÞðpðjÞÞj2 ð35Þ and Γ 2¼X j;lðjÞ;sðjÞ πpðjÞjIlðjÞðpðjÞÞj2EðjÞ¼M :ð36Þ It is important to emphasize that the above expressions are beyond a perturbative treatment of HI, see the Appendix, although the mass corrections and widths obtained are quantitatively similar to those from second order perturbation theory. Indeed, the second order perturbative correction to the energy reads δMðkÞ pert ¼X j;s Zdp jhΨðp;sÞ freeðjÞjHIjΨðkÞ boundðN−nÞij2 MðkÞ ðN−nÞ−EðpÞ ðjÞ :ð37Þ Each term in the sum over jcan be integrated separating the principal value and an imaginary contribution from the pole. Then the principal values and the pole contributions correspond respectively to the rhs of (35) and (36) with M substituted by MA¼MðkÞ ðN−nÞ. As in practice the difference induced by this substitution is quite small the second order perturbative results yield a quite good approximation. However, as shown in the Appendix, the perturbative correction at fourth order diverges, which invalidates the perturbative treatment. IV. RESULTS It is important to emphasize that our calculation of mass corrections and widths is based on the very same form of the interaction Hamiltonian we used to get the bound states. No new parameters have been introduced. Hence, the comparison to data may allow us to know the level of accuracy of our description and to think about possible improvements. A. Masses The calculated mass shifts and corrected masses for the JPC ¼1−− and ð0;1;2Þþþ states below 4.1 GeV [except the third 1−− excited state corresponding to ψð4040Þ] are presented in Table IV. Notice that we have included for completeness the first 1þþ excited state that we use for fixing the parameters of the mixing potential. We have not listed data from Xð3915Þ since its assignment to 0þþ or 2þþ is not well established. We shall come back to this state later on. A look at Table IV shows that further mass corrections have to be implemented for an accurate description of data. In this regard, we have first checked that the substitution of the nonrelativistic expression of pðjÞby the relativistic one does not make any difference. Next we analyze whether corrections to the Cornell potential could help to solve, at least in part, the observed discrepancies. To evaluate these corrections we recall that spin splittings from a single channel Cornell potential model based on (5) and (7) have been calculated [27]. Hence, for states with approximately a 100% c¯ cprobability we can just translate the results from [27]: 32.4 MeV for the 1−− ground state, −94.1MeV for the 0þþ ground state, −29.4MeV for the 1þþ ground state, and 36.5 MeV for the 2þþ ground state. This gives total masses for all these ground states differing at most 30 MeV from the experimental ones. TABLE IV. Calculated mass corrections, M −MA, and total masses, M, for JPC ¼1−− and ð0;1;2Þþþ states below 4.1 GeV. Measured masses MExpt from [1], corresponding to the meson assignment in the last column, are also given for comparison. JPC M−MA(MeV) M (MeV) MExpt (MeV) Meson 0þþ 0 3508.8 3414.71 0.3χc0ð1PÞ 2.0 3920.9 3862−32−13 þ26þ40 χc0ð3860Þ 1þþ 0 3509.8 3510.67 0.05 χc1ð1PÞ 0 3871.5 3871.69 0.17 χc1ð3872Þ 2þþ 0 3508.7 3556.17 0.07 χc2ð1PÞ −27.93881.1 3922.21.0χc2ð3930Þ −2.74003.9 1−− 0 3082.4 3096.900 0.006 J=ψð1SÞ 0 3658.8 3686.097 0.010 ψð2SÞ −14.13771.7 3778.10.7ψð3770Þ R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 074009 (2021) 074009-8 For the excited states, with significant meson-meson components, we cannot perform the same kind of analysis. Instead, spin dependent terms from the Cornell potential as well as spin dependent terms of the same order from the mixing and meson-meson potentials should be incorporated to the Schrödinger equation. As this is not an affordable task at present (we do not have any information on the spin dependence of the mixing potential) we just point out that the reasonable values of the masses we get for the excited states, as compared to the ones from experimental candidates, and the values of the spin splittings from the single channel Cornell potential model [27,28] suggest that the spin corrections to the masses, apart from the ones related to the use of the experimental meson masses, should be at most of a few tens of MeV. This makes us confident of our prediction of a second excited 2þþ state with a mass about 4 GeV. B. Widths The calculated widths for JPC ¼1−− and ð0;1;2Þþþ states below 4.1 GeV and above the D¯ Dthreshold [except the third 1−− excited state corresponding to ψð4040Þ] are compiled and compared to existing data in Table V. A look at it shows that the results for the total widths are fairly good. So, for ψð3770Þthe calculated ΓD¯ Dis about a 10% below the experimental interval (we have used that ΓExpt D¯ D ΓExpt total ¼0.93þ8 −9for a better comparison), and for χc2ð3930Þ about a 30% above data [for χc0ð3860Þbetter data are needed for a sensible comparison]. Moreover, we predict for χc2ð3930Þa dominant decay to D¯ D, as experimentally observed. We can go further since we know for χc2ð3930Þ the experimental ratio ΓExpt D¯ DΓExpt γγ ΓExpt total ¼0.21 0.04 keV:ð38Þ We can calculate this ratio from ΓD¯ Dand ΓTheor total in Table V, and Γðχc2ð3930Þ→γγÞ, which can be estimated from [29] as Γðχc2ð3930Þ→γγÞ Γðχc2ð1PÞ→γγÞ≈jðu0 c¯ cð0ÞÞχc2ð3930Þj2 jðu0 c¯ cð0ÞÞχc2ð1PÞj2¼0.80;ð39Þ where u0 c¯ cð0Þstands for the derivative of the c¯ cradial wave function at the origin. Then, using the measured value Γðχc2ð1PÞ→γγÞExpt ¼0.57 0.05 keV ð40Þ we get ΓTheor D¯ DΓTheor γγ ΓTheor total ≈0.45 0.04 keV ð41Þ which is a factor 2 bigger than data. These results for χc2ð3930Þseem to indicate that we overestimate the probability of the c¯ ccomponent since a (by hand) modest decrease in it makes the total width as well as the calculated ratio to be close to data. This overestimation may have to do with the previously mentioned need of refining our approach for a correct description of ψð4040Þ. This refinement should also affect our predicted width for the second excited 2þþ state and first excited 0þþ state. Anyhow, we think we can safely predict in both cases widths of at most a few tens of MeV. It is also worth commenting about the Xð3915Þ.In Ref. [1] D0¯ D0appears as a possible decay mode. If confirmed Xð3915Þwould be a 2þþ resonance since this decay mode is forbidden for 0þþ. The 2þþ assignment would mean its identification with χc2ð3930Þ. This is perfectly compatible with the average measured masses, 3918.41.9MeV for Xð3915Þand 3922.21.0MeV for χc2ð3930Þ. Moreover, according to our theoretical description of χc2ð3930Þthe discovery channel for Xð3915Þ, J=ψω, would be a natural decay mode of χc2ð3930Þfrom its D¯ Dcomponent via quark exchange. Hence, the confirmation of the experimental observation of the D¯ D decay mode for Xð3915Þcould definitely settle the question about its JPC quantum numbers and its possible identification with χc2ð3930Þ. To conclude this section, let us point out that the effects of open-charm meson-meson on charmonium properties has been extensively investigated in the literature, see for example [26,28,30–37]. Although our diabatic results for mass corrections and widths cannot be directly compared to the ones calculated within different frameworks, some TABLE V. Calculated widths to open-charm meson-meson channels. Experimental total widths ΓExpt total from [1], corresponding to the meson assignment in the last column, are quoted for comparison. JPC M (MeV) ΓD¯ D(MeV) ΓD¯ D(MeV) ΓDs¯ Ds(MeV) ΓD¯ D(MeV) ΓTheor total (MeV) ΓExpt total (MeV) Meson 0þþ 3920.9 0.6 0.6 201þ154þ88 −67−82 χc0ð3860Þ 1þþ 3871.7 0 <1.2χc1ð3872Þ 2þþ 3881.1 49.5 0.4 49.9 35.32.8χc2ð3930Þ 4003.9 4.8 6.3 3.5 14.5 1−− 3771.7 20.2 20.2 27.21.0ψð3770Þ DIABATIC …II. MASS CORRECTIONS …PHYS. REV. D 103, 074009 (2021) 074009-9